Runtime Complexity (innermost) proof of /tmp/tmpjePpHb/Ex24_GM04

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtUsableRulesProof (⇔, 0 ms)

↳6 CdtProblem

↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CdtProblem

↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CdtProblem

↳11 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 171 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 280 ms)

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 CdtProblem

↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳22 CdtProblem

↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳24 CdtProblem

↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳28 CdtProblem

↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 313 ms)

↳30 CdtProblem

↳31 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 237 ms)

↳32 CdtProblem

↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳34 CdtProblem

↳35 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳36 CdtProblem

↳37 CdtUnreachableProof (⇔, 0 ms)

↳38 CdtProblem

↳39 CdtUsableRulesProof (⇔, 0 ms)

↳40 CdtProblem

↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)), 0 ms)

↳42 CdtProblem

↳43 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 1 ms)

↳44 CdtProblem

↳45 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 0 ms)

↳46 CdtProblem

↳47 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳48 BOUNDS(1, 1)

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(f(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
ACTIVE(g(b)) → c2
ACTIVE(b) → c3
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(b) → c9
PROPER(c) → c10
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
ACTIVE(g(b)) → c2
ACTIVE(b) → c3
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(b) → c9
PROPER(c) → c10
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

PROPER(c) → c10
ACTIVE(g(b)) → c2
PROPER(b) → c9
ACTIVE(b) → c3

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, g, proper, f, top

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c11, c12, c13

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c11, c12, c13

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(g(z0)) → c4(G(active(z0)), ACTIVE(z0)) by

ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b)))
ACTIVE(g(b)) → c4(G(mark(c)), ACTIVE(b))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b)))
ACTIVE(g(b)) → c4(G(mark(c)), ACTIVE(b))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b)))
ACTIVE(g(b)) → c4(G(mark(c)), ACTIVE(b))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))

K tuples:none
Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b)))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(f(z0, g(z0), z1))) → c4(G(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c4(G(mark(c)), ACTIVE(g(b)))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))

K tuples:none
Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4, c4

(11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

K tuples:none
Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4, c4, c2

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(g(b)) → mark(c)
active(b) → mark(c)
proper(b) → ok(b)
proper(g(z0)) → g(proper(z0))
proper(c) → ok(c)
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
active(g(z0)) → g(active(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))

And the Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = [2]x₁
POL(F(x₁, x₂, x₃)) = 0
POL(G(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = 0
POL(b) = [2]
POL(c) = 0
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(x₁, x₂)) = x₁ + x₂
POL(f(x₁, x₂, x₃)) = 0
POL(g(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4, c4, c2

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(g(b)) → mark(c)
active(b) → mark(c)
proper(b) → ok(b)
proper(g(z0)) → g(proper(z0))
proper(c) → ok(c)
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
active(g(z0)) → g(active(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))

And the Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = [2]x₁
POL(F(x₁, x₂, x₃)) = 0
POL(G(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = 0
POL(b) = [1]
POL(c) = 0
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(x₁, x₂)) = x₁ + x₂
POL(f(x₁, x₂, x₃)) = 0
POL(g(x₁)) = [4]x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c7, c8, c11, c12, c13, c4, c4, c2

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(f(z0, z1, z2)) → c7(F(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1), PROPER(b))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(b), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(b), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c8, c11, c12, c13, c4, c4, c2, c7

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, PROPER, F, TOP

Compound Symbols:

c1, c5, c6, c8, c11, c12, c13, c4, c4, c2, c7, c7

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)), PROPER(b))
PROPER(g(c)) → c8(G(ok(c)), PROPER(c))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)), PROPER(b))
PROPER(g(c)) → c8(G(ok(c)), PROPER(c))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)), PROPER(b))
PROPER(g(c)) → c8(G(ok(c)), PROPER(c))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c12, c13, c4, c4, c2, c7, c7, c8

(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c12, c13, c4, c4, c2, c7, c7, c8, c8

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c12(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c12(TOP(ok(c)), PROPER(c))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c12(TOP(ok(c)), PROPER(c))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)), PROPER(b))
TOP(mark(c)) → c12(TOP(ok(c)), PROPER(c))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12, c12

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(b)) → c12(TOP(ok(b)))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(g(b)) → mark(c)
active(b) → mark(c)
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
active(g(z0)) → g(active(z0))

And the Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = 0
POL(F(x₁, x₂, x₃)) = 0
POL(G(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = 0
POL(b) = [4]
POL(c) = 0
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(f(x₁, x₂, x₃)) = 0
POL(g(x₁)) = 0
POL(mark(x₁)) = x₁
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(c)) → c12(TOP(ok(c)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
TOP(mark(b)) → c12(TOP(ok(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12, c12

(31) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(c)) → c12(TOP(ok(c)))

We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(g(b)) → mark(c)
active(b) → mark(c)
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
active(g(z0)) → g(active(z0))

And the Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = 0
POL(F(x₁, x₂, x₃)) = 0
POL(G(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(b) = [4]
POL(c) = 0
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(f(x₁, x₂, x₃)) = [4]
POL(g(x₁)) = [4]
POL(mark(x₁)) = [4]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, TOP, PROPER

Compound Symbols:

c1, c5, c6, c11, c13, c4, c4, c2, c7, c7, c8, c8, c12, c12

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c13(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b)))
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b))
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b)))
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b))
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b)))
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b))
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c11, c4, c4, c2, c7, c7, c8, c8, c12, c12, c13

(35) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

TOP(mark(b)) → c12(TOP(ok(b)))
TOP(mark(c)) → c12(TOP(ok(c)))
TOP(ok(b)) → c13(TOP(mark(c)), ACTIVE(b))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b)))
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))

S tuples:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b)))
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))

K tuples:

ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))

Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

ACTIVE, G, F, PROPER, TOP

Compound Symbols:

c1, c5, c6, c11, c4, c4, c2, c7, c7, c8, c8, c12, c13

(37) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(f(z0, g(z0), z1)) → c1(F(z1, z1, z1))
ACTIVE(g(g(z0))) → c4(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(b)) → c4(G(mark(c)))
ACTIVE(g(f(z0, g(z0), z1))) → c2(G(mark(f(z1, z1, z1))))
ACTIVE(g(f(z0, g(z0), z1))) → c2(ACTIVE(f(z0, g(z0), z1)))
ACTIVE(g(g(b))) → c2(G(mark(c)))
ACTIVE(g(g(b))) → c2(ACTIVE(g(b)))
PROPER(f(x0, x1, f(z0, z1, z2))) → c7(F(proper(x0), proper(x1), f(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(f(z0, z1, z2)))
PROPER(f(x0, x1, g(z0))) → c7(F(proper(x0), proper(x1), g(proper(z0))), PROPER(x0), PROPER(x1), PROPER(g(z0)))
PROPER(f(x0, f(z0, z1, z2), x2)) → c7(F(proper(x0), f(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(f(z0, z1, z2)), PROPER(x2))
PROPER(f(x0, g(z0), x2)) → c7(F(proper(x0), g(proper(z0)), proper(x2)), PROPER(x0), PROPER(g(z0)), PROPER(x2))
PROPER(f(f(z0, z1, z2), x1, x2)) → c7(F(f(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(f(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(f(g(z0), x1, x2)) → c7(F(g(proper(z0)), proper(x1), proper(x2)), PROPER(g(z0)), PROPER(x1), PROPER(x2))
PROPER(f(x0, x1, b)) → c7(F(proper(x0), proper(x1), ok(b)), PROPER(x0), PROPER(x1))
PROPER(f(x0, x1, c)) → c7(F(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(f(x0, b, x2)) → c7(F(proper(x0), ok(b), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(x0, c, x2)) → c7(F(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(f(b, x1, x2)) → c7(F(ok(b), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(f(c, x1, x2)) → c7(F(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(g(f(z0, z1, z2))) → c8(G(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(b)) → c8(G(ok(b)))
PROPER(g(c)) → c8(G(ok(c)))
TOP(mark(f(z0, z1, z2))) → c12(TOP(f(proper(z0), proper(z1), proper(z2))), PROPER(f(z0, z1, z2)))
TOP(mark(g(z0))) → c12(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(z0, g(z0), z1))) → c13(TOP(mark(f(z1, z1, z1))), ACTIVE(f(z0, g(z0), z1)))
TOP(ok(g(b))) → c13(TOP(mark(c)), ACTIVE(g(b)))
TOP(ok(g(z0))) → c13(TOP(g(active(z0))), ACTIVE(g(z0)))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

S tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

active, f, g, proper

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

(39) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(z0)) → g(active(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1, z2)) → f(proper(z0), proper(z1), proper(z2))
proper(g(z0)) → g(proper(z0))
proper(b) → ok(b)
proper(c) → ok(c)

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

S tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

G(mark(z0)) → c5(G(z0))

We considered the (Usable) Rules:none
And the Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x₁, x₂, x₃)) = x₂² + x₁³
POL(G(x₁)) = x₁³
POL(c11(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

S tuples:

G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

K tuples:

G(mark(z0)) → c5(G(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

(43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x₁, x₂, x₃)) = [5]x₁ + [3]x₂ + [3]x₃
POL(G(x₁)) = 0
POL(c11(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [4] + x₁

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

S tuples:

G(ok(z0)) → c6(G(z0))

K tuples:

G(mark(z0)) → c5(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

G(ok(z0)) → c6(G(z0))

We considered the (Usable) Rules:none
And the Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x₁, x₂, x₃)) = x₁ + [3]x₃ + x₁·x₃ + [2]x₁·x₂ + [3]x₂²
POL(G(x₁)) = [2]x₁
POL(c11(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))

S tuples:none
K tuples:

G(mark(z0)) → c5(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c11(F(z0, z1, z2))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c5, c6, c11

(47) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(0) Obligation:

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

(2) Obligation:

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(4) Obligation:

(5) CdtUsableRulesProof (EQUIVALENT transformation)

(6) Obligation:

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(10) Obligation:

(11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

(12) Obligation:

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(14) Obligation:

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(16) Obligation:

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(18) Obligation:

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(20) Obligation:

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(22) Obligation:

(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(24) Obligation:

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(26) Obligation:

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(28) Obligation:

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(30) Obligation:

(31) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(32) Obligation:

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(34) Obligation:

(35) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(36) Obligation:

(37) CdtUnreachableProof (EQUIVALENT transformation)

(38) Obligation:

(39) CdtUsableRulesProof (EQUIVALENT transformation)

(40) Obligation:

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)) transformation)

(42) Obligation:

(43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(44) Obligation:

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

(46) Obligation:

(47) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

(48) BOUNDS(1, 1)